Hopf Bifurcations in a Watt Governor With a Spring

This paper pursues the study carried out by the authors in "Stability and Hopf bifurcation in a hexagonal governor system", focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor differential system. Here are studied the codimension two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, ilustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found an open region in the parameter space where two attracting periodic orbits coexist with an attracting equilibrium point.Comment: 30 pages and 7 figure

Curvatures of conflict surfaces in Euclidean 3-space

This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane established by Siersma In the present case a conflict surface arises equidistant from the given convex sets The Gaussian Mean Curvatures and the location of Umbilic Points on the conflict surface are determined here Initial results on the Darbouxian type of Umbilic Points on conict surfaces are also established The results are expressed in terms of the principal directions and on the curvatures of the borders of the given convex set

Structural Determinants of Adhesion by Protocadherin-19 and Implications for Its Role in Epilepsy

Non-clustered δ-protocadherins are homophilic cell adhesion molecules essential for the development of the vertebrate nervous system, as several are closely linked to neurodevelopmental disorders. Mutations in protocadherin-19 (PCDH19) result in a female-limited, infant-onset form of epilepsy (PCDH19-FE). Over 100 mutations in PCDH19 have been identified in patients with PCDH19-FE, about half of which are missense mutations in the adhesive extracellular domain. Neither the mechanism of homophilic adhesion by PCDH19, nor the biochemical effects of missense mutations are understood. Here we present a crystallographic structure of the minimal adhesive fragment of the zebrafish Pcdh19 extracellular domain. This structure reveals the adhesive interface for Pcdh19, which is broadly relevant to both non-clustered δ and clustered protocadherin subfamilies. In addition, we show that several PCDH19-FE missense mutations localize to the adhesive interface and abolish Pcdh19 adhesion in in vitro assays, thus revealing the biochemical basis of their pathogenic effects during brain development

Mechanical oscillations in lasing microspheres

We investigate the feasibility of activating coherent mechanical oscillations in lasing microspheres by modulating the laser emission at a mechanical eigenfrequency. To this aim, 1.5% Nd3+:Barium-Titanium-Silicate microspheres with diameters around 50 {\mu}m were used as high quality factor (Q>10^6) whispering gallery mode lasing cavities. We have implemented a pump-and-probe technique in which the pump laser used to excite the Nd3+ ions is focused on a single microsphere with a microscope objective and a probe laser excites a specific optical mode with the evanescent field of a tapered fibre. The studied microspheres show monomode and multi-mode lasing action, which can be modulated in the best case up to 10 MHz. We have optically transduced thermally-activated mechanical eigenmodes appearing in the 50-70 MHz range, the frequency of which decreases with increasing the size of the microspheres. In a pump-and-probe configuration we observed modulation of the probe signal up to the maximum pump modulation frequency of our experimental setup, i.e., 20 MHz. This modulation decreases with frequency and is unrelated to lasing emission, pump scattering or thermal effects. We associate this effect to free-carrier-dispersion induced by multiphoton pump light absorption. On the other hand, we conclude that, in our current experimental conditions, it was not possible to resonantly excite the mechanical modes. Finally, we discuss on how to overcome these limitations by increasing the modulation frequency of the lasing emission and decreasing the frequency of the mechanical eigenmodes displaying a strong degree of optomechanical coupling.Comment: 17 pages, 5 figure

Group Strategyproof Pareto-Stable Marriage with Indifferences via the Generalized Assignment Game

We study the variant of the stable marriage problem in which the preferences of the agents are allowed to include indifferences. We present a mechanism for producing Pareto-stable matchings in stable marriage markets with indifferences that is group strategyproof for one side of the market. Our key technique involves modeling the stable marriage market as a generalized assignment game. We also show that our mechanism can be implemented efficiently. These results can be extended to the college admissions problem with indifferences

Stable schedule matching under revealed preference

Baiou and Balinski (Math. Oper. Res., 27 (2002) 485) studied schedule matching where one determines the partnerships that form and how much time they spend together, under the assumption that each agent has a ranking on all potential partners. Here we study schedule matching under more general preferences that extend the substitutable preferences in Roth (Econometrica 52 (1984) 47) by an extension of the revealed preference approach in Alkan (Econom. Theory 19 (2002) 737). We give a generalization of the GaleShapley algorithm and show that some familiar properties of ordinary stable matchings continue to hold. Our main result is that, when preferences satisfy an additional property called size monotonicity, stable matchings are a lattice under the joint preferences of all agents on each side and have other interesting structural properties

High frequency mechanical excitation of a silicon nanostring with piezoelectric aluminum nitride layers

A strong trend for quantum based technologies and applications follows the avenue of combining different platforms to exploit their complementary technological and functional advantages. Micro and nano-mechanical devices are particularly suitable for hybrid integration due to the easiness of fabrication at multi-scales and their pervasive coupling with electrons and photons. Here, we report on a nanomechanical technological platform where a silicon chip is combined with an aluminum nitride layer. Exploiting the AlN piezoelectricity, Surface Acoustic Waves are injected in the Si layer where the material has been localy patterned and etched to form a suspended nanostring. Characterizing the nanostring vertical displacement induced by the SAW, we found an external excitation peak efficiency in excess of 500 pm/V at 1 GHz mechanical frequency. Exploiting the long term expertise in silicon photonic and electronic devices as well as the SAW robustness and versatility, our technological platform represents a strong candidate for hybrid quantum systems

Anderson Photon-Phonon Colocalization in Certain Random Superlattices

International audienceFundamental observations in physics ranging from gravitational wave detection to laser cooling of a nanomechanical oscillator into its quantum ground state rely on the interaction between the optical and the mechanical degrees of freedom. A key parameter to engineer this interaction is the spatial overlap between the two fields, optimized in carefully designed resonators on a case-by-case basis. Disorder is an alternative strategy to confine light and sound at the nanoscale. However, it lacks an a priori mechanism guaranteeing a high degree of colocalization due to the inherently complex nature of the underlying interference processes. Here, we propose a way to address this challenge by using GaAs=AlAs vertical distributed Bragg reflectors with embedded geometrical disorder. Because of a remarkable coincidence in the physical parameters governing light and motion propagation in these two materials, the equations for both longitudinal acoustic waves and normal-incidence light become practically equivalent for excitations of the same wavelength. This guarantees spatial overlap between the electromagnetic and displacement fields of specific photon-phonon pairs, leading to strong light-matter interaction. In particular, a statistical enhancement in the vacuum optomechanical coupling rate, g o , is found, making this system a promising candidate to explore Anderson localization of high frequency (∼20 GHz) phonons enabled by cavity optomechanics. The colocalization effect shown here unlocks the access to unexplored localization phenomena and the engineering of light-matter interactions mediated by Anderson-localized states